Green’s function

Some general preliminary considerations

Let $(\mathrm{\Omega },\mu )$ be a bounded measure space and $ℱ(\mathrm{\Omega })$ be a linear functionspace of bounded functions defined on $\mathrm{\Omega }$, i.e. $ℱ(\mathrm{\Omega })\subset {ℒ}^{\mathrm{\infty }}(\mathrm{\Omega })$.We would like to note two types of functionals from the dual space ${(ℱ(\mathrm{\Omega }))}^{*}$, whichwill be used here:

1. Each function $g(x)\in {ℒ}^1(\mathrm{\Omega })$ defines a functional $\phi \in {(ℱ(\mathrm{\Omega }))}^{*}$ in thefollowing way:

   $$\phi (f)=\underset{\mathrm{\Omega }}{\int }g(x)f(x)𝑑\mu .$$

   Such functional we will call regular functional and function $g$ — its generator.

2. For each $x\in \mathrm{\Omega }$, we will consider a functional ${\delta }_x\in {(ℱ(\mathrm{\Omega }))}^{*}$ defined as follows:

   $${\delta }_x(f)=f(x).$$(1)

   Since generally, we can not speak about values at the point for functions from ${(L)}^{\mathrm{\infty }}$,in the following, we assume some regularity for functions from considered spaces, so that(1) is correctly defined.

Necessary notations and motivation

Let $({\mathrm{\Omega }}_x,{\mu }_x),({\mathrm{\Omega }}_y,{\mu }_y)$ be some bounded measure spaces; $ℱ({\mathrm{\Omega }}_x),𝒢({\mathrm{\Omega }}_y)$ be somelinear function spaces. Let $A:ℱ({\mathrm{\Omega }}_x)\to 𝒢({\mathrm{\Omega }}_y)$ be a linear operator which has a well-definedinverse $A^{-1}:𝒢({\mathrm{\Omega }}_y)\to ℱ({\mathrm{\Omega }}_x)$.

Consider an operator equation:

$$Af=g$$(2)

where $f\in ℱ({\mathrm{\Omega }}_x)$ is unknown and $g\in 𝒢({\mathrm{\Omega }}_y)$ is given. We are interested to have an integral representationfor solution of (2). For this purpose we write:

$$f(x)={\delta }_x(f)={\delta }_x(A^{-1}(g))=[{(A^{-1})}^{*}{\delta }_x](g).$$

Definition of Green’s function

If $\forall x\in {\mathrm{\Omega }}_x$ the functional ${(A^{-1})}^{*}{\delta }_x$ is regular with generator$G(\cdot ,y)\in {ℒ}^1({\mathrm{\Omega }}_y)$, then $G$ is called Green’s function ofoperator $A$ and solution of (2) admits the following integral representation:

$$f(x)=\underset{{\mathrm{\Omega }}_y}{\int }G(x,y)g(y)𝑑{\mu }_y$$